To study the associated Higgs boson production, we consider in this paper
the minimal \(U(1)_X\)  extension of the SM [1]}, [2]},
where the particle content is the same as the minimal \(B-L\)  (baryon minus lepton number) model [3]}, [4]}, [5]}, [6]}, [7]}
while the \(U(1)_X\)  charge of each SM particle is defined as a linear combination of its \(U(1)_Y\)  hyper-charge and \(U(1)_{B-L}\)  charge.
In the presence of three right-handed neutrinos (RHSs), this model is free from all the gauge and the mixed gauge-gravitational anomalies.
After the breaking of \(U(1)_X\)  and electroweak symmetries, the seesaw mechanism works to generate light neutrino masses naturally.
The minimal \(U(1)_X\)  model provides interesting high energy collider phenomenologies.
For the current LHC bound from the search for the \(Z^\prime \)  boson resonance with dilepton final states,
see, for example, [8]}, [9]}, [10]}.
A suitable choice of the \(U(1)_X\)  charges leads to a significant enhancement of the RHN pair production cross section
from the \(Z^\prime \)  boson which can increase the potential of the RHN discovery at the future high energy colliders
through the prompt RHN decay [11]}, [12]}, [13]}, [14]}
and the displaced vertex search for a long-lived RHN [15]}, [16]}, [17]}.
The main point of this paper is to show interesting effects of the \(Z^\prime \)  mediated process
on the associated Higgs boson production at high energy colliders.
